Equilibria, Fixed Points, and Complexity Classes

Abstract

Many models from a variety of areas involve the computation of an
equilibrium or fixed point of some kind. Examples include Nash
equilibria in games; market equilibria; computing optimal strategies
and the values of competitive games (stochastic and other games);
stable configurations of neural networks; analysing basic stochastic
models for evolution like branching processes and for language like
stochastic context-free grammars; and models that incorporate the
basic primitives of probability and recursion like recursive Markov
chains. It is not known whether these problems can be solved in
polynomial time. There are certain common computational principles
underlying different types of equilibria, which are captured by the
complexity classes PLS, PPAD, and FIXP. Representative complete
problems for these classes are respectively, pure Nash equilibria in
games where they are guaranteed to exist, (mixed) Nash equilibria in
2-player normal form games, and (mixed) Nash equilibria in normal
form games with 3 (or more) players. This paper reviews the
underlying computational principles and the corresponding classes.